For this type of random events you have to chase the robber and get back the stolen item(which is just a wallet in most of the cases). As you said, you should just chase them on a car and run him over. If he didn't drop a wallet(the dot on the radar is still red, pointing on the guy), pull back a few meters and drive him over again. Repeat until he drops the wallet.
You could also step out of your car(after you knocked him down the first time) and beat/shoot him. I once had a robber that dropped a wallet after he heard/felt some gunshots screaming something like Ok, ok, whatever, take it!
If the robber is using a bike/car, try chasing him and shooting. Aim for the robber(fairly difficult) or for the wheels(easier). If you manage to pop the wheel(s), the car will stop and the robber will most probably get out and try to shoot you.
Once you get the wallet your balance will go up(I've had amounts between $200 and $2000). Then you have a choice whether to give the wallet/item back to the victim or keep it. The first time I encountered such an event, I got a tooltip saying that I can return it for some money(which is obviously less than was in the wallet) and get a chance to increase one of the stats or get some other bonus. Or I could just keep the wallet to myself. I've always been returning it and never got any non-monetary bonus yet. The victims thank you by taking only 90% of found sum.
So, for example if you got a wallet with $500 in it and you return it, the citizen will take $450 of it(returning $50 of the $500 to you as a thank). The ratio is always the same.
And what I did as well: right after returning the money to the victim I shot him/her and got the remaining sum(or maybe a little more) back.
So for these events I'd return everything hoping for some reward and then just kill the person and get the money back.
There are also other random events, I don't know them all, I only encountered robbery and car hijacking. However I wasn't able to keep up with the bad guy and he drove away(back then I was playing as Trevor with only like 30% driving skill).
Best Answer
You can call 911 (located in your contacts).
This is the easiest way to get hold of police cars, firetrucks and ambulances.